Birational invariants, purity and the Gersten conjecture Lectures at ...

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Note that the functor X ↦−→ H 0 (X, H i (µ ⊗j
n )) is a natural contravariant functor on the
category of all k-varieties. If we restrict this functor to the category of smooth integral
)) by the group appearing in b), which is in
the notation of § 2 is none other than F loc (X) for the functor F (A) = H i (A, µ ⊗j
n ), we
k-varieties, and we replace H 0 (X, H i (µ ⊗j
n
get another proof of Prop. 2.1.10. Note however that the proof of Prop. 2.1.10 only used
the specialization property, whereas the present one uses the codimension one purity
theorem, hence the Gersten conjecture.
Remark 4.1.3 : The k-birational invariance of the groups H 0 (X, H i (µ ⊗j
n )) on smooth,
proper, integral varieties, was also observed (in zero characteristic) by Barbieri-Viale
[BV92a]. M. Rost tells me that the k-birational invariance of the group of elements in
Theorem 4.1.1.a can be proved in a “field-theoretic” fashion, without using the Gersten
conjecture (see [Ro93]).
PROPOSITION 4.1.4. — Let k be a field and n > 0 be an integer prime to char.(k).
Let i, j and m > 0 be integers. Let K = k(t) be the rational field in one variable over
k. Then the natural map H i (k, µ ⊗j
n ) → H i (k(t), µ ⊗j
n ) induces a bijection H i (k, µ ⊗j
n ) ≃
Hnr(k(t)/k, i µ ⊗j
n ).
Proof : First assume that k is perfect. For the affine line A 1 k
, and for a closed, reduced,
proper subset F ⊂ A 1 k
, we may write the long exact sequence of cohomology with support
and use purity (§ 3.3 or § 3.4) to translate it as
. . . → H i (A 1 k, µ ⊗j
n
) → H i (A 1 k − F, µ ⊗j
n ) → ⊕ H i−1 (k(P ), µ n ⊗j−1 ) → . . .
P ∈F
Now homotopy invariance for étale cohomology (see § 3.1) guarantees that the pull-back
map H i (k, µ ⊗j
n ) → H i (A 1 k , µ⊗j n ) induced by the structural morphism is an isomorphism.
For any F , the map H i (k, µ ⊗j
n ) → H i (A 1 k
− F, µ⊗j n ) is injective, as may be seen by
specializing to a k-rational point if k is infinite, or by using a 0-cycle of degree one and a
norm argument if k is finite (as a matter of fact, H i (k, µ ⊗j
n ) = 0 for k finite and i > 1).
We thus get short exact sequences
0 → H i (k, µ ⊗j
n
) → H i (A 1 k − F, µ ⊗j
n ) → ⊕ H i−1 (k(P ), µ n ⊗j−1 ) → 0.
P ∈F
We may now let F be bigger and bigger, and we ultimately get the exact sequence
(4.1) 0 → H i (k, µ ⊗j
n ) → H i (k(t), µ ⊗j
n ) → ⊕
P ∈A 1(1)
k
H i−1 (k(P ), µ ⊗j−1
n ) → 0
from which the proposition follows. (If k is not perfect, simply observe that étale
cohomology does not change by purely inseparable extensions.)